Maximizing the validity of the Gaussian Approximation for the biphoton State from parametric down-conversion Baghdasar Baghdasaryan1 2Fabian Steinlechner3 4yand Stephan Fritzsche1 2 4

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Maximizing the validity of the Gaussian Approximation for the biphoton State from

parametric down-conversion

Baghdasar Baghdasaryan,1, 2, ∗Fabian Steinlechner,3, 4, †and Stephan Fritzsche1, 2, 4

1Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨at Jena, 07743 Jena, Germany

2Helmholtz-Institut Jena, 07743 Jena, Germany

3Fraunhofer Institute for Applied Optics and Precision Engineering IOF, 07745 Jena, Germany

4Abbe Center of Photonics, Friedrich-Schiller-University Jena, 07745 Jena, Germany

(Dated: December 20, 2022)

Spontaneous parametric down-conversion (SPDC) is widely used in quantum applications based

on photonic entanglement. The eﬃciency of photon pair generation is often characterized by means

of a sinc(L∆k/2) function, where Lis the length of the nonlinear medium and ∆kis the phase

mismatch between the pump and down-converted ﬁelds. In theoretical investigations, the sinc

behavior of the phase mismatch has often been approximated by a Gaussian function exp (−αx2)

in order to derive analytical expressions for the SPDC process. Diﬀerent values have been chosen

in the literature for the optimization factor α, for instance, by comparing the widths of sinc and

Gaussian functions or the momentum of down-converted photons. As a consequence, diﬀerent values

of αprovide diﬀerent theoretical predictions for the same setup. Therefore an informed and unique

choice of this parameter is necessary. In this paper, we present a choice of αwhich maximizes the

validity of the Gaussian approximation. Moreover, we also discuss the so-called super -Gaussian

and cosine-Gaussian approximations as practical alternatives with improved predictive power for

experiments.

I. INTRODUCTION

In spontaneous parametric down-conversion (SPDC),

a nonlinear-responding quadratic crystal is pumped by a

laser ﬁeld, in order to convert (high-energy) photons into

entangled photon pairs. Entangled states generated by

SPDC have provided an experimental platform for funda-

mental quantum technologies, such as quantum cryptog-

raphy [1], quantum teleportation [2], or optical quantum

information processing [3], and by including recent mile-

stone experiments in photonic quantum computing [4].

The quantum theory of down-converted pairs is well

known and plays a crucial role in the modeling of SPDC

experiments. The state of such a SPDC pair is also

known as a biphoton state. Within the paraxial ap-

proximation, where the longitudinal and transversal com-

ponents of the wave vector are treated separately, k=

q+kz(ω)z, the biphoton state can be written as [5–7]

|Ψi=NZZ dqsdqidωsdωi

pump

z }| {

V(qs+qi) Sp(ωs+ωi)

×sincL∆kz

2

| {z }

phase matching

ˆa†

s(qs, ωs) ˆa†

i(qi, ωi)|vaci.

(1)

In expression (1), Nis the normalization factor, V(qp)

is the spatial and Sp(ωp) the spectral distribution of the

pump beam, Lis the length of the nonlinear crystal, |vaci

∗baghdasar.baghdasaryan@uni-jena.de

†fabian.steinlechner@uni-jena.de

is the vacuum state, ωs,i and qs,i are the energies and

transverse components of wave vectors of down-converted

ﬁelds (signal and idler), and ˆa†

s,i(qs,i) are the correspond-

ing creation operators.

Expression (1) can be simpliﬁed if we approximate

the sinc function in terms of the Gaussian function

sinc(x2)≈exp(−x2). Many useful analytical expressions

can be then deﬁned for SPDC within this approxima-

tion. A good example is the expression for the Schmidt

number of the biphoton state presented in Ref. [8].

Already in Ref. [8], the calculations within the Gaus-

sian approximation delivered a small deviation from ex-

perimentally measured values for the Schmidt number. It

has often been suggested in the literature to optimize this

approximation by presenting an optimization factor αin

the exponential expression exp (−αx2). The value of α

has been chosen, for example, by matching the widths of

Gaussian and sinc functions [9–11], by matching second-

order momenta [12], by comparing momentum correla-

tions in SPDC [13], or by comparing the coincidence and

single-particle spectral widths of the biphoton state [14].

Obviously, diﬀerent values of αdeliver diﬀerent theo-

retical predictions for the same setup. A unique choice

of α, which should minimize the error of the approxima-

tion, is lacking. In order to solve this problem, we look in

this paper at the states themselves, instead of comparing

just the sinc and Gaussian functions or other observable.

Eventually, the goal is to make the distance between sinc-

and Gaussian-like states as small as possible. We use an

appropriate distance measure for this purpose such as ﬁ-

delity [15] and ﬁnd the particular αthat maximizes it.

Except for the Gaussian approximation, we will also dis-

cuss the cosine- and super -Gaussian approximations and

will compare them with the usual Gaussian approxima-

tion. (All these functions can be compared in Fig. 3,

arXiv:2210.02340v2 [quant-ph] 19 Dec 2022

where we already included the optimized values of α.)

II. THEORY AND RESULTS

We ﬁrst should determine the pump and phase-

matching characteristics from expression (1), in order

to calculate the ﬁdelity of the sinc- and Gaussian-like

states. The pump beam is usually ﬁxed by the ex-

perimental setup, and its function in expression (1) is

well known. The phase mismatch in the zdirection

∆kz=kp,z −ks,z −ki,z requires more careful determi-

nation, which depends on many characteristics of the ex-

periment, such as the crystal type, the polarization of

interacting beams, and the geometry of the setup. The

derivation of ∆kzhas already been reported for a very

general experimental scenario in Ref. [7]. In comparison

to Ref. [7], here, we additionally assume degenerate mo-

mentum vectors for signal and idler photons kp≈2ks,

which allows us to apply the Gaussian approximation.

With this in mind, the phase mismatch ∆kzcan be writ-

ten as [7]

∆kz=Ωs+ Ωi

up−Ωs

us−Ωi

+|qs−qi|2

2kp

,(2)

where Ωjis the deviation from the central frequency ωj,0,

ωj=ωj,0+ Ωjwith the assumption Ωjωj,0, and

uj= 1/(∂kj/∂Ω) is the group velocity evaluated at the

central frequency.

The right-hand side of Eq. (2) can be divided into two

parts. The ﬁrst three terms represent the spectral and

the last term the spatial properties of the biphoton state.

The spectral and spatial degrees of freedom (DOFs) are

in general coupled due to the phase-matching character-

istics in SPDC [7,16–18]. However, under certain ap-

proximations, such as the narrowband [19], thin-crystal

[20,21], or plane wave approximations [10], either only

the ﬁrst three terms survive (frequency-resolved bipho-

ton state) or only the last one survives (spatially resolved

biphoton state). In this paper, we ﬁrst develop the Gaus-

sian approximation for spectral and spatial DOFs sepa-

rately. The description of the coupling in spectral and

spatial domains is more challenging in the scope of the

Gaussian approximation, which is also discussed in this

paper.

A. Spatially resolved biphoton state

The signal and idler ﬁelds can be assumed to be

monochromatic if narrowband ﬁlters are used in front

of the detectors. This is called the narrowband approxi-

mation, which transforms the biphoton state into

|Ψi=NZZ dqsdqiV(qs+qi) sincL|qs−qi|2

4kp

×ˆa†

s(qs) ˆa†

i(qi)|vaci.(3)

Expression (3) has now a simple form so that we can

apply the Gaussian, super -Gaussian, or cosine-Gaussian

approximations and calculate the corresponding ﬁdelity.

We will present the detailed calculation only for the

Gaussian approximation since the derivations for the su-

per -Gaussian and cosine-Gaussian approximations are

similar.

1. Gaussian approximation

Similar to expression (1), the approximated Gaus-

sianstate is written as

|ΨGi=NGZZ dqsdqiV(qs+qi) exp−αL|qs−qi|2

4kp

×ˆa†

s(qs) ˆa†

i(qi)|vaci,(4)

where NGis the new normalization constant. The ﬁdelity

of the states from Eqs. (3) and (4), which is simply the

scalar product of these states, is then given by

hΨ|ΨGi=N·NGZZ dqsdqiV(qs+qi)2

×sincL|qs−qi|2

4kpexp−αL|qs−qi|2

4kp.

(5)

The integral is diﬃcult to calculate in the momentum

space but mathematically more straightforward in the

position space. We use the notations q−=qs−qiand

q+=qs+qiand rewrite the function from Eq. (5) with

their Fourier transforms

hΨ|ΨGi=N·NGZZ dqsdqi

2πZdrΦF(r)eirq−

×1

2πZdr

0VF(r

0)eir

q+,

where the transformed functions are given by

ΦF(r) = 1

2πZdq−sincL|q−|2

4kpexp−αL|q−|2

4kp

×e−irq−(6)

and

VF(r

0) = 1

2πZdq+V(q+)2e−ir

q+.(7)

The integrals over momentum space can now be imple-

mented

hΨ|ΨGi=N·NGZZ drdr

0ΦF(r) VF(r

×1

2πZdqseiqs(r+r

0)1

2πZdqie−iqi(r−r

0),

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MaximizingthevalidityoftheGaussianApproximationforthebiphotonStatefromparametricdown-conversionBaghdasarBaghdasaryan,1,2,FabianSteinlechner,3,4,yandStephanFritzsche1,2,41Theoretisch-PhysikalischesInstitut,Friedrich-Schiller-UniversitatJena,07743Jena,Germany2Helmholtz-InstitutJena,07743Jena,Germany...

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Maximizing the validity of the Gaussian Approximation for the biphoton State from parametric down-conversion Baghdasar Baghdasaryan1 2Fabian Steinlechner3 4yand Stephan Fritzsche1 2 4

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